What is a PRP ?
A PRP is a probable prime number, a number that nobody knows how to prove
or disprove its primality.
Here are some criterions that can be used to recognize these numbers
:
- N does not have any prime factor up to 2^32.
- N cannot be trivially written into a product.
- Take 3 square-free consecutive bases such that (b1/N)=(b2/N)=(b3/N)= -1. N
should be such that b1^((N-1)/2)=b2^((N-1)/2)=b3^((N-1)/2)= -1 (mod N).
- Take a prime base such that (b/N)= -1 and b<>b1<>b2<>b3.
N should pass the strong test in base b.
- or more (if someone has got an idea) !!!
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Rank |
Probable prime |
Digits (> 50000) |
Who ? |
When ? |
9001 | 6^97783+77 | 76090 | Lelio R Paula | 01/2015 |
9002 | (5*10^76083-179)/3 | 76084 | Robert Price | 10/2020 |
9003 | (2^252716+1842)/2 | 76075 | Stefano Morozzi | 01/2020 |
9004 | 2^252712-105 | 76074 | Lelio R Paula | 03/2012 |
9005 | (10^76069-19)/9 | 76069 | Robert Price | 10/2017 |
9008 | 131^35927+2 | 76068 | Lelio R Paula | 05/2013 |
9006 | (41*10^76067+373)/9 | 76068 | Robert Price | 04/2017 |
9007 | (10807^18859-1)/10806 | 76068 | Serge Batalov | 08/2020 |
9009 | 2^252683-139 | 76066 | Lelio R Paula | 03/2012 |
9010 | 137^35597-136 | 76061 | Norbert Schneider | 05/2015 |
9011 | 3^159413+106 | 76060 | Lelio R Paula | 01/2015 |
9012 | (2^252593+1682)/2 | 76038 | Stefano Morozzi | 01/2020 |
9013 | 9^79677+3*8^79677-2*4^79677-3*2^79677 | 76032 | Mike Oakes | 12/2010 |
9014 | (6716^19867-1)/6715 | 76030 | Serge Batalov | 08/2020 |
9015 | 22235^2626+2626^22235 | 76029 | Hans Havermann | 12/2019 |
9016 | (2^252557-9)/71 | 76026 | Lelio R Paula | 07/2012 |
9018 | (85*10^76021-103)/9 | 76022 | Robert Price | 10/2017 |
9017 | (2^252539+2690)/2 | 76022 | Stefano Morozzi | 01/2020 |
9019 | 11^72989+80 | 76011 | Lelio R Paula | 02/2015 |
9020 | 99877!6-36 | 75996 | Robert Price | 11/2018 |
9021 | 2^252449-459523 | 75995 | Enzo Doria | 05/2015 |
9022 | 20^58408+73 | 75991 | Lelio R Paula | 01/2015 |
9023 | 2^252418-1421 | 75986 | Yves Minet | 01/2013 |
9024 | (11*2^252396-1)/5 | 75980 | Lelio R Paula | 02/2014 |
9025 | 20871^4366+4366^20871 | 75973 | Hans Havermann | 01/2020 |
9026 | 2^252365-110689 | 75970 | Enzo Doria | 05/2014 |
9027 | 11^72949-1020 | 75969 | Enzo Doria | 11/2012 |
9028 | (2^252335+2394)/2 | 75961 | Stefano Morozzi | 01/2020 |
9029 | (2^252294+47)/6549 | 75945 | Lelio R Paula | 07/2012 |
9030 | (166*10^75941-7)/3 | 75943 | Robert Price | 11/2020 |
9031 | 2^252261-110595 | 75939 | Enzo Doria | 05/2014 |
9032 | 7^89853+76 | 75935 | Lelio R Paula | 01/2015 |
9033 | (3043^21799-1)/3042 | 75930 | Serge Batalov | 08/2020 |
9034 | 3^159132+56 | 75926 | Lelio R Paula | 01/2015 |
9035 | 730^26513+26513^730 | 75916 | Norbert Schneider | 11/2016 |
9036 | 7*10^75911-57 | 75912 | Robert Price | 06/2020 |
9037 | (27*2^252163-1)/1945 | 75907 | Lelio R Paula | 01/2014 |
9038 | 5^108596-14846 | 75906 | Enzo Doria | 02/2013 |
9039 | (1377^24179-1)/1376 | 75894 | Serge Batalov | 07/2020 |
9040 | (2^252082+2734)/2 | 75884 | Stefano Morozzi | 01/2020 |
9041 | 5^108560-14846 | 75881 | Enzo Doria | 02/2013 |
9042 | 2^252051-110359 | 75875 | Enzo Doria | 05/2014 |
9043 | 2^251971-110509 | 75851 | Enzo Doria | 05/2014 |
9044 | (42457^16384+1)/327682 | 75819 | Lelio R Paula | 11/2011 |
9045 | 2^251842-4715 | 75812 | Yves Minet | 05/2019 |
9046 | 99655!6-2 | 75811 | Robert Price | 07/2017 |
9047 | (2^251823+3290)/2 | 75806 | Stefano Morozzi | 01/2020 |
9048 | 14*10^75798+1089 | 75800 | Jens K. Andersen | 11/2007 |
9049 | 2^251784-455 | 75795 | Yves Minet | 03/2012 |
9050 | 2^251780-1289 | 75794 | Yves Minet | 01/2013 |
9051 | 2^251760-215609 | 75788 | Paul Bourdelais | 03/2013 |
9052 | 21947^2838+2838^21947 | 75784 | Hans Havermann | 01/2020 |
9053 | 99^37973+2 | 75781 | Lelio R Paula | 05/2013 |
9054 | (185*10^75777+7)/3 | 75779 | Robert Price | 02/2017 |
9055 | 5*2^251690+3 | 75767 | Lelio R Paula | 06/2014 |
9056 | 6^97366+95 | 75766 | Lelio R Paula | 01/2015 |
9057 | (4742^20611-1)/4741 | 75762 | Serge Batalov | 08/2020 |
9058 | s(23010) | 75760 | Hans Havermann | 11/2017 |
9059 | (2^251666+1882)/2 | 75759 | Stefano Morozzi | 01/2020 |
9060 | 8*6^97337-7 | 75744 | Lelio R Paula | 10/2014 |
9061 | 24045^1412+1412^24045 | 75738 | Hans Havermann | 01/2020 |
9062 | 2^251588-111677 | 75736 | Enzo Doria | 05/2014 |
9063 | (2^251580+2086)/2 | 75733 | Stefano Morozzi | 01/2020 |
9064 | 5^108346-13256 | 75731 | Enzo Doria | 02/2013 |
9065 | (2^251570+1182)/2 | 75730 | Stefano Morozzi | 01/2020 |
9066 | (31^50777+1)/32 | 75726 | Robert Price | 07/2013 |
9067 | (10^75707+77)/3 | 75707 | Robert Price | 01/2017 |
9068 | (174907#+1673)/(7*239*15467899) | 75700 | Makoto Morimoto | 10/2017 |
9069 | 2^251466+193 | 75699 | Lelio R Paula | 03/2012 |
9070 | (2^251463+1338)/2 | 75698 | Stefano Morozzi | 01/2020 |
9071 | (5*2^251464+1)/9423 | 75695 | Lelio R Paula | 07/2012 |
9072 | 2^251422-109991 | 75686 | Enzo Doria | 04/2014 |
9073 | 91*(10^23+1)*(10^1721+1)*(10^118749+1)/(10^69+1)/(10^5163+1)/(10^39583+1) | 75680 | Makoto Kamada | 03/2019 |
9074 | (2^251355+3806)/2 | 75666 | Stefano Morozzi | 01/2020 |
9075 | (2^251347+1278)/2 | 75663 | Stefano Morozzi | 01/2020 |
9076 | 4*5^108245-3 | 75661 | Lelio R Paula | 10/2014 |
9077 | (25*10^75653+143)/3 | 75654 | Robert Price | 08/2020 |
9078 | (2^251261+846)/2 | 75637 | Stefano Morozzi | 01/2020 |
9079 | (2^251195+3770)/2 | 75617 | Stefano Morozzi | 01/2020 |
9080 | 12^70067-11^70067 | 75615 | Jean-Louis Charton | 11/2006 |
9081 | 12^70066+95 | 75614 | Lelio R Paula | 02/2015 |
9082 | (2^251181+1130)/2 | 75613 | Stefano Morozzi | 01/2020 |
9083 | -E20534/2063 | 75607 | Thomas Ritschel | 02/2017 |
9084 | 19^59120-50 | 75600 | Lelio R Paula | 01/2015 |
9085 | 18^60222-4697 | 75596 | Enzo Doria | 03/2013 |
9086 | 18^60211-4355 | 75582 | Enzo Doria | 03/2013 |
9087 | 7*2^251014-9 | 75564 | Lelio R Paula | 06/2014 |
9088 | 6^97101-25 | 75560 | Lelio R Paula | 01/2015 |
9089 | 8^83663-7^83663 | 75556 | Jean-Louis Charton | 01/2004 |
9090 | (2^250974+3838)/2 | 75551 | Stefano Morozzi | 01/2020 |
9091 | 3^158331-140 | 75544 | Lelio R Paula | 01/2015 |
9092 | (12235^18481-1)/12234 | 75539 | Serge Batalov | 08/2020 |
9093 | (29*10^75532-107)/3 | 75533 | Robert Price | 08/2020 |
9096 | (37^48157-1)/36 | 75519 | Robert Price | 03/2012 |
9094 | (19*10^75518-43)/3 | 75519 | Robert Price | 10/2017 |
9095 | 2^250868-4229 | 75519 | Yves Minet | 08/2018 |
9097 | 10^75516+31467 | 75517 | Patrick De Geest | 03/2016 |
9099 | 10^75516-10^2256-1 | 75516 | Serge Batalov | 01/2011 |
9098 | 10^75516-66591 | 75516 | Patrick De Geest | 03/2016 |
9100 | 2^250851-187 | 75514 | Lelio R Paula | 03/2012 |
9101 | (2^250836+466)/2 | 75509 | Stefano Morozzi | 01/2020 |
9102 | 22273^2454+2454^22273 | 75503 | Hans Havermann | 01/2020 |
9103 | (43*10^75498-7)/9 | 75499 | Robert Price | 12/2014 |
9104 | (3^158244-5)/247916 | 75497 | Lelio R Paula | 06/2012 |
9105 | 4*5^107971-3 | 75470 | Lelio R Paula | 10/2014 |
9106 | (25*10^75465+71)/3 | 75466 | Robert Price | 01/2017 |
9107 | (2^250690+1222)/2 | 75465 | Stefano Morozzi | 01/2020 |
9109 | (5*2^250681-1)/999 | 75461 | Lelio R Paula | 01/2014 |
9108 | (2^250674+1354)/2 | 75461 | Stefano Morozzi | 01/2020 |
9110 | (2^250670+1878)/2 | 75459 | Stefano Morozzi | 01/2020 |
9111 | (182*10^75456+1)/3 | 75458 | Robert Price | 02/2017 |
9112 | (2^250669-9)/98897729 | 75451 | Lelio R Paula | 07/2012 |
9113 | (14583^18121-1)/14582 | 75449 | Serge Batalov | 08/2020 |
9114 | 10^75441+13 | 75442 | Robert Price | 01/2011 |
9115 | 10^75441-9923 | 75441 | Patrick De Geest | 05/2013 |
9116 | (27*2^250546-1)/6457 | 75420 | Lelio R Paula | 01/2014 |
9117 | 5*7^89231-6 | 75410 | Lelio R Paula | 10/2014 |
9119 | 2^250503-16995 | 75409 | Lelio R Paula | 05/2013 |
9118 | 11^72411-28 | 75409 | Lelio R Paula | 02/2015 |
9120 | 3*5^107860-8 | 75392 | Lelio R Paula | 10/2014 |
9121 | (34*10^75387-403)/9 | 75388 | Robert Price | 10/2017 |
9124 | 2^250426+29835 | 75386 | Lelio R Paula | 05/2013 |
9122 | 8*3^157999-7 | 75386 | Lelio R Paula | 05/2014 |
9123 | (15671^17971-1)/15670 | 75386 | Serge Batalov | 08/2020 |
9125 | (2^250417+2642)/2 | 75383 | Stefano Morozzi | 01/2020 |
9126 | (88*10^75381-43)/9 | 75382 | Robert Price | 11/2015 |
9127 | (2^250394+2902)/2 | 75376 | Stefano Morozzi | 01/2020 |
9128 | 2^250382-111755 | 75373 | Enzo Doria | 05/2014 |
9129 | 2^250374-2798051 | 75371 | Eric Chen | 03/2016 |
9130 | (22*10^75368-1)/3 | 75369 | Robert Price | 12/2014 |
9131 | 2^250369-4543 | 75369 | Yves Minet | 03/2019 |
9132 | 20078^5667+5667^20078 | 75360 | Hans Havermann | 02/2020 |
9133 | (2^250318+2242)/2 | 75353 | Stefano Morozzi | 01/2020 |
9134 | (21398^17401-1)/21397 | 75349 | Serge Batalov | 08/2020 |
9136 | (19*10^75338+119)/3 | 75339 | Robert Price | 02/2017 |
9135 | 21517^3172+3172^21517 | 75339 | Hans Havermann | 01/2020 |
9137 | 2^250223-1859167 | 75325 | Enzo Doria | 03/2016 |
9138 | (3^157854-7)/66025282 | 75308 | Lelio R Paula | 06/2012 |
9139 | (2^250164+1578)/2 | 75307 | Stefano Morozzi | 01/2020 |
9140 | 3^157831-98 | 75305 | Lelio R Paula | 01/2015 |
9141 | (46^45281+11^45281)/57 | 75290 | Serge Batalov | 12/2016 |
9142 | 92^38335-91 | 75282 | Norbert Schneider | 01/2016 |
9143 | (2^250026+2398)/2 | 75266 | Stefano Morozzi | 01/2020 |
9144 | 2^250020+18915 | 75264 | Lelio R Paula | 05/2013 |
9145 | 2^250002+8193 | 75259 | Lelio R Paula | 05/2013 |
9146 | (2^249996+1678)/2 | 75256 | Stefano Morozzi | 01/2020 |
9147 | (11*2^249977-7)/15 | 75251 | Renaud Lifchitz | 02/2011 |
9148 | (2^249968+78557)/63 | 75247 | Lelio R Paula | 07/2012 |
9149 | (5*10^75244+91)/3 | 75245 | Robert Price | 11/2016 |
9150 | (10489!)^2+Prime(10489) | 75242 | Robert Price | 08/2014 |
9151 | 14^65644+67 | 75237 | Lelio R Paula | 01/2015 |
9152 | 68383!4+32 | 75236 | Robert Price | 09/2019 |
9153 | (2^249928-5)/11859059 | 75229 | Lelio R Paula | 06/2012 |
9154 | 129^35638+2 | 75218 | Lelio R Paula | 05/2013 |
9155 | 2^249853-110475 | 75214 | Enzo Doria | 05/2014 |
9156 | (5704^20021-1)/5703 | 75199 | Serge Batalov | 08/2020 |
9157 | 20574^4517+4517^20574 | 75195 | Hans Havermann | 01/2020 |
9158 | (11720^18481-1)/11719 | 75194 | Serge Batalov | 08/2020 |
9159 | 2^249762+75409 | 75186 | Enzo Doria | 04/2016 |
9160 | 565^27318+27318^565 | 75181 | Norbert Schneider | 12/2016 |
9161 | (2^249741-7)/685 | 75177 | Lelio R Paula | 06/2012 |
9162 | 2^249726-112181 | 75176 | Enzo Doria | 06/2014 |
9163 | 2^249721+11715 | 75174 | Lelio R Paula | 05/2013 |
9164 | (2^249705+654)/2 | 75169 | Stefano Morozzi | 01/2020 |
9166 | (38661^16384+1)/2 | 75158 | Lelio R Paula | 11/2011 |
9165 | (15220^17971-1)/15219 | 75158 | Serge Batalov | 08/2020 |
9167 | 2^249658-47355 | 75155 | Lelio R Paula | 05/2013 |
9168 | 155^34311+2 | 75153 | Lelio R Paula | 05/2013 |
9169 | (10^87682+1)/(10^12526+1)/990099009901 | 75145 | Maksym Voznyy | 06/2015 |
9170 | (23*2^249602+1)/93 | 75138 | Lelio R Paula | 08/2012 |
9171 | 2^249522+8415 | 75114 | Lelio R Paula | 05/2013 |
9172 | 2^249489-110911 | 75104 | Enzo Doria | 05/2014 |
9173 | (28*10^75078+131)/3 | 75079 | Robert Price | 02/2017 |
9174 | (206*10^75073-17)/9 | 75075 | Robert Price | 03/2017 |
9175 | (2^249384+3330)/2 | 75072 | Stefano Morozzi | 01/2020 |
9176 | (5947^19891-1)/5946 | 75071 | Serge Batalov | 08/2020 |
9177 | 2^249368-110573 | 75068 | Enzo Doria | 05/2014 |
9178 | NaturalLogarithmOf10(75067) | 75067 | Eric W. Weisstein | 10/2015 |
9179 | 19167^8248+8248^19167 | 75065 | Hans Havermann | 02/2020 |
9180 | (2^249375+78557)/45056125 | 75062 | Lelio R Paula | 07/2012 |
9181 | (26*10^75055-173)/3 | 75056 | Robert Price | 10/2017 |
9183 | 2^249272+193 | 75039 | Lelio R Paula | 03/2012 |
9182 | 17433^17692-17692^17433 | 75039 | Norbert Schneider | 11/2019 |
9184 | 14^65468+43 | 75035 | Lelio R Paula | 01/2015 |
9185 | 37^47844+36 | 75030 | Norbert Schneider | 05/2014 |
9186 | (52*10^75027-7)/9 | 75028 | Robert Price | 12/2014 |
9187 | 17943^15178+15178^17943 | 75024 | Hans Havermann | 02/2020 |
9188 | (2^249219+494)/2 | 75023 | Stefano Morozzi | 01/2020 |
9189 | 20^57653+89 | 75009 | Lelio R Paula | 01/2015 |
9190 | (2^249084+2682)/2 | 74982 | Stefano Morozzi | 01/2020 |
9191 | 2^249061-1868353 | 74975 | Enzo Doria | 04/2016 |
9192 | (2^249056+1006)/2 | 74974 | Stefano Morozzi | 01/2020 |
9193 | 7^88713+2*3^88713 | 74972 | Mike Oakes | 07/2010 |
9194 | 18325^12334+12334^18325 | 74970 | Hans Havermann | 02/2020 |
9195 | 2*6^96313+5 | 74947 | Lelio R Paula | 10/2014 |
9197 | (8*10^74944+49)/3 | 74945 | Robert Price | 03/2016 |
9196 | (1258^24179-1)/1257 | 74945 | Serge Batalov | 07/2020 |
9198 | 300^30247+30247^300 | 74926 | Norbert Schneider | 11/2014 |
9199 | 2^248866-171 | 74917 | Lelio R Paula | 03/2012 |
9200 | (714^26251-1)/713 | 74910 | Serge Batalov | 08/2020 |
9201 | 6*5^107168-7 | 74908 | Lelio R Paula | 10/2014 |
9202 | 11^71926+78 | 74904 | Lelio R Paula | 02/2015 |
9203 | 2^248824+1869391 | 74904 | Enzo Doria | 05/2016 |
9204 | 2^248816+1111 | 74902 | Lelio R Paula | 01/2011 |
9205 | V(388219) | 74890 | Serge Batalov | 03/2015 |
9206 | 159^34017+158 | 74885 | Norbert Schneider | 12/2018 |
9207 | 2*F(358297)-1 | 74880 | Henri Lifchitz | 12/2009 |
9208 | (22*10^74874-31)/9 | 74875 | Robert Price | 04/2015 |
9209 | 11^71892-76800000 | 74868 | Enzo Doria | 01/2013 |
9210 | 9*5^107107-4 | 74866 | Lelio R Paula | 10/2014 |
9211 | 2^248678-710111 | 74860 | Paul Bourdelais | 03/2013 |
9212 | (14*10^74858+211)/9 | 74859 | Robert Price | 11/2017 |
9213 | 13^67194+12 | 74851 | Lelio R Paula | 01/2015 |
9214 | (7*2^248634+1)/71 | 74846 | Lelio R Paula | 08/2012 |
9215 | 11^71867-3090 | 74842 | Enzo Doria | 12/2012 |
9216 | 68057!4-16 | 74841 | Robert Price | 11/2019 |
9217 | 2^248609+262145 | 74839 | Lelio R Paula | 05/2013 |
9218 | 3^156852+50 | 74838 | Lelio R Paula | 01/2015 |
9219 | 18^59607-11 | 74824 | Lelio R Paula | 01/2015 |
9220 | 2^248556-110507 | 74823 | Enzo Doria | 05/2014 |
9221 | 7^88531-36 | 74818 | Lelio R Paula | 01/2015 |
9222 | p(15738)#/2+1024 | 74814 | Robert Price | 07/2017 |
9223 | (2^248505+422)/2 | 74808 | Stefano Morozzi | 01/2020 |
9224 | (2^248507+1938)/2 | 74808 | Stefano Morozzi | 01/2020 |
9225 | 21137^3454+3454^21137 | 74790 | Hans Havermann | 11/2019 |
9226 | 4*3^156740-5 | 74785 | Lelio R Paula | 05/2014 |
9227 | 17918^14907+14907^17918 | 74779 | Hans Havermann | 12/2019 |
9228 | (32*10^74773-41)/9 | 74774 | Robert Price | 05/2015 |
9229 | (29*2^248384-1)/13 | 74772 | Lelio R Paula | 01/2014 |
9230 | 19401!+19401#+1 | 74765 | D. Gazzoni Filho & Lucas Teixeira | 06/2005 |
9231 | 2^248357-12797413 | 74763 | Enzo Doria | 03/2016 |
9232 | 42^46057+41 | 74762 | Lelio R Paula | 07/2012 |
9233 | (2^248339+1686)/2 | 74758 | Stefano Morozzi | 01/2020 |
9234 | 11^71784-3000 | 74756 | Enzo Doria | 12/2012 |
9235 | 19^58458+18 | 74754 | Lelio R Paula | 01/2015 |
9236 | 12^69265-35 | 74750 | Lelio R Paula | 02/2015 |
9237 | (2^248291+830)/2 | 74743 | Stefano Morozzi | 01/2020 |
9239 | 11^71771-48000 | 74742 | Enzo Doria | 12/2012 |
9238 | (7*10^74741+197)/3 | 74742 | Robert Price | 07/2017 |
9240 | (2^248282-5)/50129 | 74736 | Lelio R Paula | 06/2012 |
9241 | 2^248252-110105 | 74732 | Enzo Doria | 04/2014 |
9242 | (7*10^74729+71)/3 | 74730 | Robert Price | 03/2016 |
9243 | 3^156614-50 | 74724 | Lelio R Paula | 01/2015 |
9244 | (7*2^248213+1)/135 | 74719 | Lelio R Paula | 07/2012 |
9245 | (2^248193+870)/2 | 74714 | Stefano Morozzi | 01/2020 |
9246 | (2^248167+1178)/2 | 74706 | Stefano Morozzi | 01/2020 |
9247 | (2^248173+78557)/767 | 74705 | Lelio R Paula | 07/2012 |
9248 | (2^248145+3866)/2 | 74699 | Stefano Morozzi | 01/2020 |
9249 | 125^35622+124 | 74697 | Norbert Schneider | 02/2015 |
9250 | (2^248114+994)/2 | 74690 | Stefano Morozzi | 01/2020 |